Please Use Python codes to solve this problem. Thank you!

There is a measurement station on the roof of the building you work in that measures wind

direction and the concentration of Nitrogen $($C$)$ in the air. You know from many years of

measurements that when winds are from the east $(E),$ the air that arrives at the measurement

instrument has anomalous Nitrogen concentrations that follow a normal distribution with

mean ${}_E=0\mathrm{.}2$ and St$.$ deviation ${}_E=1($note that these are not estimates, these are the true

values of the distribution$).$ On the other hand, winds from the west $($W$)$ bring relatively pristine

air that has anomalous Nitrogen concentrations that follow a normal distribution with mean

${}_W=0\mathrm{.}0$ and St$.$ deviation ${}_W=1.$

But there is a problem. You just went to check on the instrument, and you found that it is no

longer recording the wind direction or the concentrations! Due to a glitch in the software, all

you know is that the average Nitrogen concentration of the air over the past $100$ samples was

$\frac{?}{\text{b}\text{a}\text{r}}(C{)}_{100}=0\mathrm{.}13.$ Do you think the wind over the last $100$ samples was from the east or the west?

$($assume that the wind did not change over the past $100$ samples, i$.$e$.$ it was either consistently

from the east or consistently from the west$).$

Q$1.$ Use hypothesis testing: can you reject the null hypothesis that the air in the last $100$

samples is actually air from the west $(H0$ : The air is from the west$)?$ Use a two$-$tailed test with

confidence level of $0\mathrm{.}05$ and describe your steps.

Q$2.$ You start discussing the issue with your boss. After analyzing the wind rose information,

you see that the winds blow out of the west $($between SW and NW$)40\%$ of the time and that the

winds blow out of the east $($between NE and SE$)60\%$ of the time. Let $$ denote the fraction of

time the air comes from the west $($so$,Y=0\mathrm{.}4).$ Use Bayes' Theorem to compute the probability

that the air came from the west over the last $100-$sample period.

Q$3.$ You will now explore the sensitivity of your results to the parameter $.$ That is$,$ you want

to understand what would happen if your climatological wind fraction was different $($say$,$ if

you repeated this analysis in a different location$).$ Thus, your final result will be a single plot

showing how often the z$-$statistic from Part $($a$)$ guesses the wind direction correctly and how

often the Bayes formulation from Part $($b$)$ guesses the wind direction correctly as a function

of $,$ where $$ takes on values between $0$ and $1.$ Explain your results and what they imply about

the two different methods. Be sure to specifically discuss the behavior in the limits of large

and small $.$

Hints $($For Q$3)$: To set up this problem, you will design and run Monte Carlo experiments to

simulate this physical situation. For each $in[0,0\mathrm{.}1,0\mathrm{.}2,0\mathrm{.}3,0\mathrm{.}4,0\mathrm{.}5,0\mathrm{.}6,0\mathrm{.}7,0\mathrm{.}8,0\mathrm{.}9,1],$ write

code to:

$(1)$ Randomly choose which scenario to run according to the probability $Y,($generate a

random number from $U(0,1)$ : if it is $Y,$ then "winds are from the west", otherwise "winds are

from the east"$).$

$(2)$ Based on $(1),$ generate $N=100$ random concentration values from the underlying

distributions $($see problem statement$),$ and calculate their average $\frac{?}{\text{b}\text{a}\text{r}}(C{)}_N.$

$(3)$ Use the methods from $($a$)$ and $($b$)$ to calculate whether the z$-$test and Bayesian approach

conclude that the wind was from the west given the $\frac{?}{\text{b}\text{a}\text{r}}(C{)}_N$ you calculated.

$(4)$ Compare what the z$-$test and Bayesian approaches said to the truth $($the truth is the

scenario that you actually used in $(1))$ and keep track of how often each method guesses the

correct scenario.

Repeat $(1-4)$ many times $(>1000)$ for each different $Y$ value. Plot a figure of the frequency of

success of each of the two methods as a function of $.$